 Counting  Fundamental Counting principle  Factorials  Permutations and combinations  Probability  Complementary events  Compound events  Independent events  Dependent events

 Consider this:  You have an exam with five true or false questions. How many different ways are there to answer the test?  The fundamental counting principle states that: "If there are r ways to do one thing, and s ways to do another, and t ways to do a third thing, and so on..., then the number of ways of doing all those things at once is r * s * t etc...."

 You have four pairs of pants, five shirts, and six pairs of shoes. How many outfits can you make?  You are making license plates, but you are limited to letters in the first three spaces and numbers in the final three spaces. How many license plates could you make?

 A permutation is an ordering or arrangement.  Suppose that there are five students who need to ask for help on their pre-calculus homework. How many different orders could there be?  What if there were seven students instead of five?

 A more efficient way to find the products is the previous problems is factorial notation.  Factorial notation is defined as follows: For any natural number n, n! = n(n-1)(n- 2)…(2)(1) 0! = 1  Find 2!, 5!, and 5!  Use your calculator to find 13! And 20!

 There are 15 students in speech. How many arrangements are there for 3 students to give a speech?  P(n, r) represents the number of permutations for n elements, taken r at a time.  P (n, r) = n!/[(n-r)!] = n(n-1)(n-2) {for a total of r factors}  Try these:  P(7, 2)  P(10, 3)

 A combination is not a ordering or arrangement, but a subset of a set of elements.  Consider: you have 4 people in class ready to present, but you want to pick a team of two people to make a presentation. How many choices do you have?  C(n, r) represents this choice.  C(n, r) = P(n, r)/r! = n!/[(n-r)!*r!)  Try these:  C(10, 2)  C (30, 25)

 In permutations, order matters.  AB ≠ BA  In combinations, order does not matter.  AB = BA

 Probability is the likelihood that an event will occur.  It is between 0 and 1.  A probability of 0 means an event will never happen.  A probability of 1 means an event is certain to happen.  A few terms:  Outcome: the result from an experiment  Sample space: the set of all outcomes  Event: subset of sample space

 Probability is computed by dividing the favorable outcomes by possible outcomes. P(E) = n(E)/n(S) Where n(E) is the number of outcomes that are favorable and n(S) is the total outcomes  Find the probability of drawing a queen from a deck of cards.

 An event and its complement take up the entire sample space.  This is often shown as E and E’.  For example, when rolling dice, the probability of rolling a six and the probability of not rolling a six are complements. What is the probability of each?  P(E)+ P(E’) = 1

 This is the probability of two events occurring.  It is measured by the probability of each event occurring minus any overlap.  Try this one:  What is the probability of rolling two dice and getting either the sum of eight or rolling a pair?

 Suppose you toss a coin and then toss it again. What is the probability that you will get two heads?  This is the probability of independent events.  It is the probability of one event times the probability of the other.

 Imagine that you trying to find the probability of drawing two hearts from a deck of cards. In scenario 1, the card you have drawn is replaced. In scenario 2, the card is not replaced. What is the probability of each scenario?  Scenario 2 is the probability of dependent events.  This is the probability of the 1 st event and then the probability of the 2 nd event, assuming that the 1 st event happened.

 Pages #32-44 even, #56-68 even  Page 951 #22-30 even